DPRIETI Discussion Paper Series 18-E-053

Spatial Pattern and City Size Distribution(Revised)

MORI, TomoyaRIETI

The Research Institute of Economy, Trade and Industryhttps://www.rieti.go.jp/en/

RIETI Discussion Paper Series 18-E-053

First Draft: August 2018

Revised: August 2019

Spatial Pattern and City Size Distributioni

Tomoya MORI

Kyoto University, RIETI

Abstract

Many large cities are found at locations with certain first nature advantages. Yet, those exogenous

locational features may not be the most potent forces governing the spatial pattern of cities. In

particular, population size, spacing and industrial composition of cities exhibit simple, persistent and

monotonic relationships. Theories of economic agglomeration suggest that this regularity is a

consequence of interactions between endogenous agglomeration and dispersion forces. This paper

reviews the extant formal models that explain the spatial pattern together with the size distribution of

cities, and discusses the remaining research questions to be answered in this literature. To obtain results

about explicit spatial patterns of cities, a model needs to depart from the most popular two-region and

systems-of-cities frameworks in urban and regional economics in which there is no variation in

interregional distance. This is one of the major reasons that only few formal models have been

proposed in this literature. To draw implications as much as possible from the extant theories, this

review involves extensive discussions on the behavior of the many-region extension of these models.

The mechanisms that link the spatial pattern of cities and the diversity in city sizes are also discussed

in detail.

Keywords: City size distribution, Spatial patterns, Agglomeration, Racetrack geography,

Interregional distance, Power laws, Central place theory

JEL Classification: R12, C33

i This research was conducted as part of the project, “An Empirical Framework for Studying Spatial Patterns

and Causal Relationships of Economic Agglomeration,” undertaken at the Research Institute of Economy,

Trade and Industry. This research has been partially supported by the Grant in Aid for Research (Nos.

17H00987, 16K13360, 16H03613,15H03344) of the MEXT, Japan.

The RIETI Discussion Papers Series aims at widely disseminating research results in the form of

professional papers, with the goal of stimulating lively discussion. The views expressed in the papers are

solely those of the author(s), and neither represent those of the organization(s) to which the author(s)

belong(s) nor the Research Institute of Economy, Trade and Industry.

1 Introduction

In the past 50 years since the formal analyses of city formation started around the time ofAlonso (1964),1 the spatial pattern of cities has remained as a relatively minor subject inurban economics2 – despite that economic geographers in the past (e.g., von Thunen, 1826;Christaller, 1933; Losch, 1940), have commonly suggested the inseparable correspondencebetween (population) size and spatial distributions of cities (see, e.g., Fujita, 2010).3

The mainstream theories in urban economics have abstracted from the heterogeneity ininter-city/regional space by adopting the systems-of-cities model since the pioneering workby J. Vernon Henderson (Henderson, 1974),4 or by simply assuming the presence of onlytwo regions in an economy (see a collection of two-region models presented in Baldwin,Forslid, Martin, Ottaviano and Robert-Nicoud, 2003). The mechanism which determinesthe size of a city/region has always been a major subject in most of these theories, and forthis purpose, the abstraction from interregional space in these approaches substantiallysimplified the analyses.

As a consequence of this particular evolution of the field, there exist rather limitedtheoretical as well as empirical literature which relate the spatial pattern and sizes of cities.To my knowledge, there are two major strands of formal models that explicitly deal withthe spatial pattern of cities, new economic geography (NEG) and social-interactions models.The former explains city formation by the externalities that arise from monopolisticallycompetitive markets (see, e.g., Fujita et al., 1999a; Baldwin et al., 2003, for surveys),whereas the latter by the externalities that arise from direct interactions among agentsoutside the markets (see, e.g., Beckmann, 1976; Fujita and Ogawa, 1982; Fujita and Smith,1990; Mossay and Picard, 2011). This paper focuses on the basic structure and implicationsof these theoretical models in connection to the observed size and spatial patterns of cities.

In Section 2, I start by making observations on the relation among sizes, spatial patternsand industrial structure of cities in reality by using data from Japan. In Section 3, genericproperties of the canonical models (to be made precise below) of the extant theories arediscussed. In particular, while most models were investigated in the context of the two-

1There is a large literature on location theory that preceded urban economics and have importantimplications on city and agglomeration formation (see, e.g., Thisse et al., 1996, for a survey), although theywere not designed to explain city formation per se.

2A notable exceptions are Isard (1949, 1956). While no formal models have been proposed by Isard,he foresaw the necessity of increasing returns and imperfect competition in order to explain the formationof cities and their spatial pattern. In particular, he envisaged the emergence of new economic geographywhich played a central role in this literature as will be discussed in Section 3 (see Fujita, 2010, for furtherdiscussions).

3See an intriguing review by Fujita (2012) on the von Thunen’s work and ideas about spatial organizationof economy.

4See, e.g., Abdel-Rahman and Anas (2004) for a survey. See also Behrens and Robert-Nicoud (2015) formore recent applications of this framework. The standard systems-of-cities models assume zero inter-citytransport cost. There are a few variations assuming an equal distance between any pair of cities (see, e.g.,Anas and Xiong, 2003; Anas, 2004). In either case, there is only a single inter-city distance, as in the case ofa two-region model.

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region setup in their original studies, in this paper, by extensively drawing from the workof Akamatsu, Mori, Osawa and Takayama (2018), I summarize their behavior in a many-region setup in which the spatial pattern of cities can be more properly studied. Finally,Section 4 concludes the paper.

2 Facts about size, location and industrial composition of

cities

To guide summarizing and classifying the extant theoretical models for the size and spa-tial patterns of cities, it is useful to have a concrete idea about the basic relationshipsobserved between them in reality. Given that the inter-city space has been largely ab-stracted in the literature, however, systematic researches on this subject are scarce, andthe results published so far provide little decisive evidence (e.g., Dobkins and Ioannides,2001; Overman and Ioannides, 2001; Ioannides and Overman, 2004). To demonstrate thestrong correspondence between theories and facts, here, rather than trying to put togethersubtle pieces of evidences from the existing empirical literature, I attempt to develop a setof clear-cut facts using data from Japan.

There are two major reasons to focus on Japan as a real world example. One is the dataavailability of the micro data for industrial locations. The other is the fact that both thehighway and high-speed railway networks in Japan have been developed simultaneouslyalmost from scratch to the full-fledged nation-wide networks between 1970 and 2015. Thechanges in size and spatial patterns of Japanese cities in this period provides a useful testcase to verify the implications from the theoretical models of endogenous city formation.By utilizing these data, the key facts on all the aspects regarding size and spatial patternsas well as industrial structure of cities can be obtained for the same set of cities.

Throughout this section, a city is defined to be a contiguous set of (approximately)1km-by-1km cells with at least 1000 people per km2 and total population of at least10,000.5 The advantage of this simple definition of a city is that the basic regional units(1km-by-1km cells) are consistent in the cross sections of a given country, and acrossdifferent points in time, unlike more commonly used definitions of metropolitan areasbased on administrative regions. Under this definition of a city, the set of all cities ina country account for the population (area) share in the country of 43.6% (2.4%), 44.6%(1.6%), 77.1% (12.4%), 48.7% (2.9%) and 47.0% (3.8%) for the US, Europe, Japan, Chinaand India, respectively.6

5This definition of a city is a variation of that proposed by Rosenfeld et al. (2011). The results tobe presented below are not sensitive to the density and total population thresholds, unless they are setextremely high or low so that only a few high density cities or a few spatially gigantic cities are identified.

6The estimated population count data at the 1km-by-1km cell level are obtained from Statistics Bureau,Ministry of Internal Affairs and Communication of Japan (2015) for Japan, and from the LandScan by OakRidge National Laboratory (2015) for the rest of the countries.

2

It is to be noted that the evidence on Japanese cities to be presented below is not specificto Japan. For size and spatial patterns of cities discussed in Sections 2.1 and 2.2, Mori etal. (2019) have shown that qualitatively the same (and more extensive) results hold forJapan and other five countries, China, France, Germany, India and the US under the samedefinition of a city. For the size and industrial structure of cities discussed in Section 2.3,the qualitatively similar results have also been presented for the case of the US (see Hsu,2012; Schiff, 2014) under the standard metro areas and industrial classifications.

2.1 Size and spacing of cities

Many large cities are found at locations with certain first nature advantages.7 Yet, thoseexogenous locational features may not be the most potent forces governing the spatialpattern of cities. In particular, population size, spacing and industrial composition ofcities exhibit a simple, persistent and monotonic relationship, which have long beenrecognized by economic geographers since von Thunen (1826), Christaller (1933) andLosch (1940). They (especially, Christaller) suggested a central place pattern in the relationbetween the size and location of cities such that larger cities tend to serve as centers aroundwhich smaller cities are grouped. Moreover, this relation is recursive so that some of thesmall cities serve as centers around which even smaller cities are grouped. This centralplace pattern of cities naturally implies that a larger cities are more spaced apart.

To see this in the actual data, letU be the set of all 450 cities identified in Japan in 2015,si be the share of city i ∈ U in the national population, and ‖i, j‖ for i, j ∈ U be the roaddistance between cities i and j.8 Define the spacing of city i by the distance to the closestcity of the same or a larger size class:9

di = minj∈{k∈U : sk>0.75si}

‖i, j‖ . (1)

Figure 1(a) shows the relationship between di and si in log scale for each city i ∈ U. Thecorrelation between them is as high as 0.67.10 This confirms the spacing-out property ofcities mentioned above.

If the number, ni, of cities within the distance di from city i ∈U is counted by

ni ≡ #{ j ∈U\{i} : ‖i, j‖ < di} , (2)

7For the role of the natural advantage in the city formation, see, e.g., Davis and Weinstein (2002) for thecase of Japan, Bleakley and Lin (2012) and Cronon (1991) for the US, and Michaels and Rauch (2018) forFrance and the UK.

8The road distance is based on the OpenStreetMap data as of July, 2017. The distance between cities iscomputed as the distance between the centroids of the most densely populated 1km-by-1km cells in thesecities. The computation was done using the Stata interface, osrmtime, of Open Source Routing Machine byHuber and Rust (2016).

9The lower threshold share, 0.75, defining the “same size class” in (1) is arbitrary. But, the choice of thethreshold value does not alter the qualitative result as long as it is not too far from 1.0.

10The dashed line in the figure is the fitted line by Ordinary Least Squares (OLS) regression.

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as shown in Figure 1(b), it also has strong correlation, 0.86, with the city size, si, in logscale. Thus, indeed it is clear that larger cities are surrounded by smaller cities.

Mori et al. (2019) conduct a formal test for this central place pattern of cities, i.e., if thelargest cities are spaced out relative to the whole set of cities in a country. Specifically, theyfirst fix the number L of the largest cities in a given country, and form a Voronoi partitionwith respect to a set of a given number K (≥ 2) of randomly selected cities. The test statisticis the count of partition cells containing at least one of these L largest cities. If there issubstantial spacing between the largest cities in reality, then this count is expected to belarger for Voronoi partitions than for fully random partitions (i.e., without any regard tospatial relations among cities) of the same cell sizes. For a range of (L,K) values, theyfound strong evidence for the spacing-out of large cities in all the six countries considered(China, France, Germany, India, Japan and the US).11

(a) Spacing of cities in terms of distance (b) Spacing of cities in terms of city count

Figure 1: Spacing of Japanese cities in 2015

2.2 Size distribution of cities

It is well known that city size distribution in a well-urbanized country exhibits an ap-proximate power law (e.g., Gabaix and Ioannides, 2004; Batty, 2006; Bettencourt, 2013).12

Formally, if a given set of n cities is postulated to satisfy a power law, and if these citysizes are ranked as s1 ≥ s2 ≥ · · · ≥ sn, so that the rank, ri, of city i is given by ri = i, then forsome positive constants c and α,

ri/n ≈ P(S > si) ≈ cs−αi ⇒ lnsi ≈ b−1α

lnri (3)

11Dobkins and Ioannides (2001) found a negative correlation between the size and spacing of cities inthe US for the period 1900-1980. But, the specific feature of the US cities needs to be taken into account istheir historical development. The formation of cities started in the northeastern region of the US in the 19thcentury, and then expanded gradually to west and then to south. But, the effective distance kept changing inthe meantime in response to the advancement in the transport technology. As a consequence, the spacingof the same size class of cities has increased over time. Such underlying heterogeneity across regions is tosome extent taken into account in the construction of counterfactuals in the test by Mori et al. (2019).

12Dittmar (2011) shows evidence that power laws for city size distributions in Europe emerged after1500, i.e., after the dependence of city production on land relaxed substantially.

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for b = ln(cn)/α.In Figure 2, Panel (a) shows the rank-size distributions of cities in every five years

from 1970 to 2015, where si indicates the share of city i in the national population; Panel(b) shows the change in α, i.e., the Zipf’s coefficient, over these 45 years. One can seethat the city size distribution exhibits an approximate power law in each year, althoughagglomeration towards larger cities has been accelerated. The variation in city size isremarkably large, as exhibited by the largest three cities, Tokyo, Osaka and Nagoya,accounting for 45% of the total city population, where Tokyo alone accounts for 26% in2015.

There is a strand of literature which informally argue that Zipf’s law (after Zipf, 1949)holds, i.e., the power law with α = 1 holds for city size distribution in a country (see,e.g., Gabaix and Ioannides, 2004; Ioannides, 2012, §8.2). But, there is abundant evidenceagainst it (e.g., Black and Henderson, 2003; Soo, 2005; Nitsch, 2005; Mori et al., 2019), as isalso clear in the Japanese case shown in Figure 2.

While the power laws for city size distributions are best known at the country level(see, e.g., Gabaix and Ioannides, 2004), Mori et al. (2019) have shown that the similarpower laws appear recursively in spatial hierarchies of regions within a country thatreflect the central place patterns discussed in Section 2.1. Specifically, they construct aspatial hierarchy in a country by first constructing a Voronoi partition of the set of allcities in the country using a given number of their largest cities as cell centers, and thencontinuing this partitioning procedure within each cell recursively. They found that citysize distributions in different parts of these spatial hierarchies exhibit power laws that aresignificantly similar than would be expected by chance alone. Thus, their result suggeststhat city systems have a spatial fractal structure within countries.

19701975198019851990

19952000200520102015

1970 1980 1990 2000 2010 2015Year

Zipf

’s co

effici

ent

ln(Rank)

ln(P

opul

ation

shar

e)

(a) Rank-size distributions of cities (b) Change in Zipf’s coefficient

0.75

0.80

0.85

0.90

Figure 2: Rank-size distribution of cities in Japan

2.3 Size and industrial structure of cities

Many evidences (e.g. Glaeser and Mare, 2001; Bettencourt, Lobo, Helbing, Kuhnert andWest, 2007; Combes, Duranton and Gobillon, 2008; Glaeser and Resseger, 2010; Combes,Duranton, Gobillon, Puga and Roux, 2012; Baum-Snow and Paven, 2013; Davis and Dingel,

5

2019) have indicated strong correlations between socio-economic quantities and sizes ofcities (e.g., wages, education level, gross domestic product, industrial diversity, numberof patents applications, amount of crime, level of traffic congestion). This section presentsone of the clearest representations of such correlations by focusing on industrial location.

Let I be the set of all industries that operate in at least one of the cities, and for a givenindustry i ∈ I, call a city a choice city of this industry if industry i is in operation in the city.These choice cities exhibit a systematic variation in their average population size acrossindustries. To see this, denote byUi (⊆U) the set of all choice cities of industry i ∈ I, thenthe average size of choice cities for industry i is given by

si =1

#Ui

∑i∈Ui

si , (4)

where #Ui means the cardinality of setUi.Now, consider three-digit secondary and tertiary industries of the Japanese Standard

Industrial Classification (JSIC) that are present both in 2000 and 2015. Of all the 237 suchindustries, there are 162 and 175 industries that have at least one establishment in citiesin 2000 and 2015, respectively.13 Figure 3 shows the relationship between si and Ni fori ∈ I in log scale, where Ni ≡ #Ui. The dashed curves indicate the upper and lower boundfor the average size of choice cities in 2015, where for each i ∈ I, the former (latter) is theaverage size of the largest (smallest) Ni cities.

20002015

Upper bound

Lower bound

Figure 3: Varieties of economic activities and their choice of cities in Japan

There are two key features in these plots. First, the number Ni and average size si ofchoice cities exhibit a strong power law, which is persistent between 2000 and 2015. Second,the average sizes of choice cities are almost hitting their upper bound, meaning that thechoice cities of an industry i ∈ I is roughly the largest Ni cities, which in turn implies thatthere is roughly a hierarchical relationship in the industrial composition between larger andsmaller cities.14

13Data for the locations of establishments were obtained from Statistics Bureau, Ministry of InternalAffairs and Communication of Japan (2001, 2014).

14These features are first recognized by Mori et al. (2008); Mori and Smith (2011) for the case of Japan,

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To see this, let Ii represent the set of industries that are present in city i ∈ U, and forcities i and j ∈U such that si > s j, define the hierarchy share for city j with i by

Hi j =#(Ii∩I j

)#I j

∈ [0,1] , (5)

where a larger value of Hi j indicates a higher consistency with the hierarchical relationship,and Hi j = 1 means the perfect hierarchical relationship, i.e.,U j ⊆Ui. The average valuesof the hierarchy shares for all the relevant city pairs,

H ≡1

H

∑i, j∈U s.t. si>s j

Hi j ∈ [0,1] (6)

where H ≡ #{(i, j) : i, j ∈ U, si > s j}, can be taken as an aggregate measure of spatial co-ordination among industries. A larger value of H indicates a higher degree of spatialcoordination, and the coordination is perfect if H = 1. The actual values of H are 0.76 and0.80 in 2000 and 2015, respectively, which are quite high.15

Together with the central place pattern discussed above (see Figure 1), the fact that thespatial coordination of diverse economic activities leads to the diversity in city size hasalready been suggested informally by Christaller (1933) and Losch (1940).

A large value of H as in the case of Japan above means that it is not only that industrieshave different number of agglomerations (i.e., choice cities), but also that their locationstend to coincide, i.e., a more localized industry choose to locate in cities in which a moreubiquitous industries are present. The case of perfect coordination (i.e., H = 1) correspondsto the hierarchy principle in Christaller (1933).

To close this subsection, it is worth pointing out that while there is a strong tendency ofhierarchical relation in the industrial composition between a larger and a smaller cities, itis by no means the rule. Figure 4 shows the distribution of Hi j of all the relevant city pairsin 2015. While the mean value is H = 0.80, the standard deviation is 0.13, and the rangeis from 0.18 to 1. Low hierarchy shares are realized for specialized cities in which only asmall specific set of industries are concentrated. As will be discussed in Section 3.2, thestandard systems-of-cities models (e.g., Henderson, 1974; Rossi-Hansberg and Wright,2007) associate the size of a city with that in scale economies specific to the industriesin which the city is specialized, thereby explain the diversity in city size in terms of thevariation in industry-specific scale economies.

and Hsu (2012, Appendix A1) and Schiff (2014) for the case of the US. See also Davis and Dingel (2019) foran evidence of the hierarchical industrial structure of the US cities based on an alternative approach.

15These values are much higher than the values of H that can be realized under random location ofindustries after controlling for the industrial diversity (i.e., #Ii for i ∈ U) of cities and locational diversity(i.e., #Ui for i ∈ I) of industries (see, e.g., Mori et al., 2008; Mori and Smith, 2011; Mori, 2017).

7

1200010000

8000600040002000

00.2 0.4 0.6 0.8 1.0

Average : 0.80

Frequency

Figure 4: Distribution of hierarchy share between cities in Japan in 2015

2.4 Growth of city sizes

Finally, we look at the characteristics of the growth of individual city sizes in Japanbetween 1970 and 2015. It is of particular interest to quantify the evolution of city sizesin this period, since it coincides with the period in which the highway and high-speedrailway networks were developed almost from scratch to the extent that covers almostthe entire nation, where the total highway (high-speed railway) length increased from 879km (515 km) by more than 16 (10) times to 14,146 km (5,350 km).

The level of interregional transport access has been one of the key parameters to deter-mine the size and spatial patterns of cities in the literature. The evolution of the sizes ofindividual cities is expected to reflect the response to the improved interregional transportaccess, although the benefit for each city may vary depending on their relative location.Thus, the changes in size and spatial patterns experienced by Japanese cities in this periodprovides an ideal test case for the theoretical models of endogenous agglomeration.

There was substantial movement of population among cities in these 45 years. Inparticular, there is a clear tendency of global agglomeration toward a smaller number ofcities, as the number of cities has decreased from 503 to 450.16

Figure 5 reveals key facts about the change in individual city sizes for the 302 cities thathave remained throughout the entire period. Panel (a) adds another evidence for globalagglomeration: the size of the remained cities in terms of population share (in the country)has grown by 21% on average.17 Note that it is more meaningful to look at the populationshare of a city rather than the population size itself to understand the tendency of globalagglomeration, because the population shares remove the effects of general populationgrowth and/or urbanization from the population sizes.18

16Cities may emerge, disappear, split and merge over time. Cities identified in the consecutive two yearsare considered to represent the same city if they mutually account for the largest population among all theoverlapping cities.

17“S.D.” in the panels means the standard deviation.18Overman and Ioannides (2001) have shown evidence that there is mild tendency of the decrease in

population size of relatively large cities (i.e., metropolitan areas with urban core of at least 50,000 population)of the US for the period 1920-1980. Their result is not directly comparable to the case of Japan here, sincetheir results may be biased for relatively large cities, and the factors driving city sizes during the studied

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Despite the tendency of global agglomeration, there is also a clear tendency of localdispersion as the areal size of an individual city has almost doubled (Panel, b), while thepopulation density has decreased by 22% on average (Panel, c).19

This simultaneous occurrence of global agglomeration and local dispersion given animprovement in interregional access may seem paradoxical. But, it can be explained byintegrating the extant theories of endogenous agglomeration to be discussed in the nextsection.

(a) Population share (b) Area

Frequency

Mean 0.21S.D. 0.75

Mean 0.94S.D. 1.05

(c) Population densityMea -0.22S.D. 0.22

Figure 5: Changes in the sizes of individual cities in Japan between 1970 and 2015 (Growthrates in horizontarl axis)

3 Theories

A model capable of explaining the spatial patterns of cities necessarily involves manyregions with large variations in interregional distance, such that some cities are close towhile others are far from one another. But, the majority of the extant models adopt eithertwo-region20 or systems-of-cities setups21 in which there is no variation in interregionaldistance. Thus, no explicit spatial patterns reflecting the relation among the number, sizeand spacing of cities can be expressed by these models.

A recent work by Akamatsu et al. (2018) brought a breakthrough by showing that awide variety of the extant models of endogenous agglomeration can be reformulated in amany-region setup, and formally analyzed in a unified framework. Specifically, they focuson a canonical model, i.e., a static model with (i) a continuum of homogeneous mobileagents, each of whom chooses a single location; (ii) there is a single type of interregionaltransport cost; (iii) transport costs are subject to the iceberg technology. The reformulated

period were not made clear.19The suburbanization in response to the decrease in interregional transport access is one realization of

local dispersion, and its evidence for the case of the US metro areas has been reported by Baum-Snow (2007,2017). For the global agglomeration and dispersion, no clear consensus has been attained at this point inthe extant literature (e.g., Duranton and Turner, 2012; Faber, 2014; Baum-Snow, 2017). This is rather evidentfrom the discussion in Section 3 below that the effect of interregional transport access on each individual citysize is not monotonic. See Akamatsu et al. (2018, §6) for an extensive discussion on this respect. Ioannidesand Overman (2004) examined the change in the distance from each city to its nearest neighbor, and foundit was decreasing in the period of 1900 to 1990, which should essentially imply global dispersion. But, thereis no discussion on the potential causes of this change in their paper.

20See, for example, Baldwin et al. (2003) for a survey of NEG models21See, for example, Anas and Xiong (2003); Anas (2004); Tabuchi et al. (2005a).

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models are shown to boil down to one of the three reduced forms in terms of the spatialpattern of agglomeration and dispersion.

The canonical model covers a wide range of standard models in urban and regionaleconomics. It includes the class of NEG models based on the Dixit-Stiglitz type CESsubutility function for love of variety (e.g., Krugman, 1991; Helpman, 1998; Tabuchi,1998; Puga, 1999; Forslid and Ottaviano, 2003; Pfluger, 2004; Murata and Thisse, 2005;Redding and Sturm, 2008; Pfluger and Sudekum, 2008); social-interactions model of city-center formation based on technological externalities (e.g., Beckmann, 1976; Mossay andPicard, 2011; Blanchet et al., 2016); and the economic geography models in “universalgravity” framework by Allen et al. (Forthcoming), including the Armington (1969) modelwith labor mobility by Allen and Arkolakis (2014), a standard formulation in the recentquantitave spatial economics (see, e.g., Redding and Rossi-Hansberg, 2017, for a survey).

Important classes of models that are out of their scope include city-center formationmodels in which firms and households have different location incentives (violation of (i))(e.g., Fujita and Ogawa, 1982; Lucas and Rossi-Hansberg, 2002; Ahlfeldt et al., 2015; Monteet al., 2018); NEG models with additive transport costs (violation of (iii)) by Ottavianoet al. (2002), and those with multiple industries with industry-specific transport costs(violation of (ii)) (e.g., Fujita and Krugman, 1995; Fujita et al., 1999b; Tabuchi and Thisse,2011).

Drawing largely from Akamatsu et al. (2018), Section 3.1 reviews the mechanismunderlying the relation between population/areal size and spacing of cities in realitydiscussed in Sections 2.1 and 2.4. To explain the observed diversity in the size andindustrial structure of cities discussed in Sections 2.2 and 2.3, respectively, and theirrelation to the spatial pattern of cities, a model needs to go beyond the canonical modelconsidered by Akamatsu et al. (2018), and incorporate variations in the degree of increasingreturns (and/or those in transport costs). At present, there are only a handful of modelsthat have succeeded in such extensions. Section 3.2 reviews the theoretical developmentin this direction.

3.1 Spatial pattern of cities

By formalizing and generalizing the idea proposed by Krugman (1996, Ch.8) based onTuring (1952), Akamatsu, Takayama and Ikeda (2012) proposed an analytical frameworkfor many-region models of endogenous agglomeration under the symmetric racetrackgeography with the help of discrete Fourier transformation. While Akamatsu et al. (2012)has focused on a many-region extension of the model by Pfluger (2004), Akamatsu etal. (2018) have generalized their framework, and have shown that a wide variety of theextant models can be classified by the three distinct reduced forms, despite the differencein their specific mechanisms underlying agglomeration and dispersion. Below, I start bydescribing the basic setup of this approach.

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Basic setup

Consider the location space consisting of a set of K discrete regions, K ≡ {0,1, . . . ,K−1}. There is a continuum of homogeneous mobile agents whose regional distribution isdenoted by hhh ≡ (hi)i∈K , where hi is the mass of mobile agents located in region i. Theirtotal mass is a given constant, H ≡

∑i∈K hi. All regions inK are featureless and are placed

at an equal interval on a circle. In this racetrack economy, transportation is possible onlyalong the circumference.22

Let region index 0,1, . . . ,K − 1 represent the location on the racetrack in clockwisedirection. Transport costs take iceberg form, i.e., if a unit of product is shipped fromregion i to j, then only the fraction di j = d ji ∈ [0,1) reaches j. The spatial discounting matrix,DDD = [di j], expresses the underlying distance structure of the economy. Typically, icebergcosts are expressed as di j = exp[−τ`i j], where `i j is the distance between regions i and jand τ ∈ (0,∞) is the transport technology parameter.

The relocation of agents is assumed to be much slower than market reactions, sothat the short-run equilibrium conditions (such as market clearing and trade balance)determine the payoff (utility or profit) in each region as a function of a given regionaldistribution of agents, hhh. Specifically, given hhh, their short-run payoff of choosing eachregion is determined, where the short-run payoff function is denoted by vvv(hhh) ≡ (vi(hhh))i∈K ,with vi(hhh) representing the payoff for an agent located in region i ∈ K .

In the long-run, agents are mobile and are free to choose their locations to improvetheir own payoffs. In (long-run) equilibrium, it must hold that v∗ = vi(hhh) for all regions iwith hi > 0, and v∗ ≥ vi(hhh) for any region i with hi = 0, where v∗ is the equilibrium payoff

level.A change in endogenous agglomeration pattern is treated as an instance of bifurcation

of an equilibrium. To address the stability of equilibria, a standard approach in theliterature is to introduce equilibrium refinement based on local stability under myopicevolutionary dynamics, where the rate of change in the number of residents hi in regioni is modeled on the basis of the regional distribution of agents, hhh, and that of payoff,vvv(hhh). Let a deterministic dynamic be denoted by hhh = FFF(hhh,vvv(hhh)), where hhh represents thetime derivative of hhh, and assume that (i) FFF satisfies differentiability with respect to botharguments, (ii) agents relocate in the direction of an increased aggregate payoff under FFF,(iii) the total mass of agents is preserved under FFF, and (iv) any spatial equilibrium is a restpoint of the dynamic, i.e., if hhh∗ is an equilibrium, it must hold that hhh = FFF(hhh∗,vvv(hhh∗)) = 000. Thestability of an equilibrium then is defined in terms of asymptotic stability under FFF.

22The racetrack location space is obviously counterfactual, as it is edge less. Although the presence ofthe edge tends to make the agglomeration on the edge larger, since there is no competing agglomerationbeyond the edge (see, e.g., Fujita and Mori, 1997; Ikeda et al., 2017), this effect becomes negligible for a largeeconomy, and the agglomeration patterns can be approximated by that in the edge-less economy.

11

Formation of a city

With a racetrack geography, the uniform distribution of mobile agents is always an equi-librium when the payoff function is symmetric across regions. Call an equilibrium withuniform distribution a flat-earth equilibrium, and denote it by hhh ≡ (h,h, . . . ,h) with h ≡H/K.

If the adjustment dynamic is formulated so that the agents migrate in order to maxi-mize their payoff, it follows (Akamatsu et al., 2018, Appendix B) that each eigenvalue ofJacobian matrix JJJ of FFF and that of the Jacobian matrix ∇vvv of vvv are real, and have a perfectpositive correlation at the flat-earth equilibrium. Thus, one can focus on ∇vvv instead ofJJJ to investigate the stability of the flat-earth equiliburum. What remains is to identifythe direction of the bifurcation at the flat-earth equilibrium, which is equivalent to findthe eigenvector of ∇vvv(hhh) whose eigenvalue changes its sign from negative to positive firstamong all the eigenvectors of ∇vvv(hhh).

The sign of the k-th eigenvalue of ∇vvv(hhh) has been shown to coincide with the sign ofthe model-specific function of the form:

G(

fk)

= c0 + c1 fk + c2 f 2k , (7)

where c0,c1 and c2 are the constants specific to a given model, and fk is the k-th eigenvalueof the spatial discounting matrix DDD which is known to be real, and common to all models.The eigenvector associated with fk is given by ηηηk = (ηk,i) = (cos[θki]) for i ∈K withθ≡ 2π/K,and the bifurcation from the flat-earth equilibrium takes place in the direction given byhhh = hhh +εηηηk with ε > 0.

The value k coincides with the number of equidistant regions toward which mobileagents migrate the most. For example, at k = K/2, the value ηK/2,i of each element i ∈ Kin eigenvetor, ηηηK/2, is given as depicted for the case of K = 16 in Figure 6(a), so thatagglomerations start to form at alternate regions, 0,2,4, . . . ,K − 2(= 14).23 At k = 1, asdepicted in Figure 6(b), an unimodal agglomeration will form around region 0.24

(b) Monocentric pattern

0 1 3 512 1413 Region0

(a) K/2-centric pattern

Region1

23

45

67

89

1011

1213

1415

45 6 7 8 9 10 11

12

Figure 6: Agglomeration formation at high and low transport costs25

There are two key properties of f ′k s that are useful to investigate the stability of flat-earthequilibrium:

23It is equally likely that agglomerations take place at regions, 1,3, . . . ,K−1.24It is equally likely that the agglomeration takes place around any region inK .25This figure is the replication of Akamatsu et al. (2018, Figure 3).

12

1. fk is monotonically increasing in transport cost, τ.

2. f1 = maxk=1,2,...,K fk and fK/2 = mink=1,2,...,K fk > 0.26,27

Canonical models typically have a positive value of c1. Since f1 > 0, it means that thesecond term on the right hand side (RHS) in (7) represents the agglomeration force, asit works to destabilize the flat-earth equilibrium. In these models, if a stable flat-earthequilibrium exists, then one must have either c0 < 0 or c2 < 0, or both, so that all theeigenvalues of ∇vvv(hhh) can be negative at the flat-earth equilibrium. In particular, since fk ispositive and increasing in τ for each k = 1,2, . . . ,K− 1, the flat-earth equilibrium is stablefor sufficiently small transport costs if c0 < 0, and for sufficiently large transport costs ifc2 < 0.

The bifurcation from the flat-earth equilibrium leading to the city formation underc0 < 0 and that under c2 < 0 are, however, qualitatively different in two aspects. The firstaspect is the timing at which the bifurcation takes place. The bifurcation under c0 < 0takes place in the increasing phase of transport costs, whereas that under c2 < 0 in theirdecreasing phase.

The second aspect is the spatial scale of agglomeration and dispersion. Provided thatc2 < 0, the bifurcation takes place in the direction of ηK/2, i.e., every other region alongthe racetrack attracts in-migration of mobile agents, when G( fK/2) becomes positive (referto Figure 6(a)). The regional distribution of mobile agents that arises in this bifurcationis hhh + εηηηK/2 (for ε > 0) as illustrated in Figure 7(a). In other words, small cities (i.e.,agglomerations) form locally, while they are dispersed globally all over the location space.

Provided that c0 < 0, the bifurcation takes place in the direction of ηηη1 when G( f1) turnsto positive (refer to Figure 6(b)). The regional distribution of mobile agents that arises inthis case is given by hhh+εηηη1 as illustrated in Figure 7(b). In other words, the agglomerationtakes place globally, and form a single gigantic city, while the dispersion takes place locallyaround the city center (region 0) so that the city stretches over the entire location space.

Figure 7: City formation at high and low transport costs28

26 f0 whose corresponding eigenvector is η0 = (1,1, . . . ,1) is irrelevant for the stability of equilibria as thetotal mobile population is preserved.

27For simplicity, it is assumed that K is an even integer, although it is not essential.28This figure is the replication of Akamatsu et al. (2018, Figure 5).

13

A crucial difference between the two cases is the dependence of dispersion force onthe distance structure of the model. The third term c2 f 2

k on the RHS of (7) depends on thedistance structure of the economy (through fk). As discussed above, this force leads toglobal dispersion (with local agglomeration) as in Figure 7(a). On the contrary, the first termon the RHS of (7) is the dispersion force when c0 < 0 which is independent of the distancestructure of the economy. As discussed above, this force leads to local dispersion (withglobal agglomeration) as in Figure 7(b).

In Akamatsu et al. (2018), the models with only global dispersion force, i.e., c0 ≥ 0 andc2 < 0, are called Class (i). The models of this class are shown to exhibit period doublingbifurcations as transport costs decrease, leading to a smaller number of larger cities with a largerspacing between neighboring cities, until all mobile agents concentrate in one region (Figure 8a).29

The models with only local dispersion force, i.e., c0 < 0 and c2 ≥ 0 are called Class (ii). TheClass (ii) models involve at most one bifurcation when the flat-earth equilibrium loosesstability. In the models of this class, keeping unimodal regional distribution, the concentrationof mobile agents proceeds as transport costs increases, until all mobile agents concentrate in oneregion (Figure 8b). The models that incorporate both types of dispersion force, i.e., c0 < 0and c2 < 0, may be the most realistic, and account for the formation of multiple cities witha positive internal space (Figure 8c). These are called Class (iii).

larger

smaller(a) Class (i) (b) Class (ii) (c) Class (iii)

Figure 8: Spatial patterns of cities

Two implications are worth mentioning. First, the heterogeneity among interregionaldistances is an essential feature of a model to investigate the spatial pattern of cities. In thecontext of a two region model or a systems-of-cities model in which there is no variationin interregional distance, the dispersion of mobile agents in Class (i) and Class (ii) modelslook exactly the same. But, as indicated by the middle panels of Figure 8(a)(b), these arequalitatively different in spatial scale. The dispersion takes place at the global scale inClass (i) models – in the form of an increase in the number of cities, and at the local scalein Class (ii) models – in the form of a larger spatial extent of a city.

Second, the responses of agglomeration/dispersion to the level of transport costs areopposite between global and local spatial scales. More specifically, given the lower interre-gional transport costs, the agglomeration proceeds at global scale, i.e., the number of citiesdecreases, the sizes and the spacing of the remaining cities increase, while the dispersion

29See Akamatsu et al. (2012) for the formal analyses on the period doubling bifurcations of class (i)models.

14

proceeds at local scale, i.e., the average population density within a city decreases and thespatial extent of a city increases.30

Notice that the behavior of Class (i) models essentially account for the larger cities beingspaced more apart as discussed in Section 2.1, and the behavior of Class (iii) models, i.e.,the combination of Classes (i) and (ii), can account for the evolution of city growth ofJapan discussed in Section 2.4.

Below, I overview a variety of extant models that fall into one of these three classes, aswell as those do not.

New economic geography

NEG (e.g., Fujita, Krugman and Venables, 1999a) commonly utilizes the monopolisticcompetition together with scale economies in production to explain the endogenous ag-glomeration. On the one hand, the love for product variety by consumers and the presenceof transport costs give an incentive for consumers to locate closer to firms. On the otherhand, each indivisible firm subject to scale economies at the plant level has an incentiveto locate and supply near the concentration of consumers.31

In this context, the global dispersion force associated with c2 < 0 in (7) is introducedtypically by assuming immobile consumers in each region who generate dispersed de-mand for the differentiated products (e.g., Krugman, 1991, 1993; Forslid and Ottaviano,2003; Pfluger, 2004). The assumption of immobility of consumers is nothing but simpli-fication to assure the dispersed demand. It can be obtained endogenously, for example,by introducing land-intensive sectors that also require labor inputs (e.g., Fujita and Krug-man, 1995; Puga, 1999), which in turn generates dispersed demand from workers. Withtransport costs, the proximity to demand matters, and hence, the spatial dispersion ofconsumers results in the formation of multiple cities, where the firms in each city mainlyserves their nearby local market.

The local dispersion force associated with c0 < 0 in (7) is introduced by assuming con-sumption of locally scarce land (e.g., Helpman, 1998; Redding and Sturm, 2008; Reddingand Rossi-Hansberg, 2017), sometimes together with commuting costs (e.g., Murata andThisse, 2005).32 All these costs of concentration are confined within a given region, andthus are independent of interregional distance. The dispersion in this case takes the formof overflow of mobile agents from a given city to the nearby regions, rather than theformation of new distinct cities at distant regions.

30Of course, the actual evolution of the spatial patterns under the changing level of transport costs is morecomplicated, as neighboring cities may eventually merge in the case of Class (iii) models. See, Akamatsu etal. (2018, §5.3).

31An alternative formulation assumes the product variety in intermediate goods. See, e.g., Fujita et al.(1999a, Ch.14).

32A similar effect can be obtained by assuming local congestion externality that is effective within a givenregion.

15

There are models that incorporate both global and local dispersion forces above(Tabuchi, 1998; Pfluger and Sudekum, 2008), i.e., of Class (iii) with c0 < 0 and c2 < 0in (7). While these themselves treat only the two-region case, their many-region exten-sions can generate a more realistic spatial pattern of cities that involve both global andlocal dispersion as shown in Figure 8(c) (see Akamatsu et al., 2018, §5.3).33

Social interactions model

In the 1970s and 1980s, there were a series of attempts to explain endogenous formationof the central business districts (CBD) within a city. The development of the models ofthis type was initiated by Solow and Vickrey (1971) and Beckmann (1976), then followedby several others (e.g., Borukhov and Hochman, 1977; O’Hara, 1977; Ogawa and Fujita,1980; Fujita and Ogawa, 1982; Imai, 1982; Tauchen and Witte, 1983; Tabuchi, 1986; Fujita,1988; Kanemoto, 1990; Fujita, 1990).

In these models, the formation of CBD is explained by introducing positive technolog-ical externalities generated from the interaction between each pair of individual agents.While the above mentioned models vary in the specification of positive externalities, Fu-jita and Smith (1990) have shown that their formulations are essentially equivalent, andreformulated commonly by the so-called additive interaction function, Si(hhh) ≡

∑j∈K di jh j.

In the simplest specifications (as in, e.g., Beckmann, 1976), this additive interactionfunction enters the utility function of consumers directly. Most models assume land con-sumption by mobile agents, while the production sector is abstracted, i.e., they incorporateonly local dispersion force, and hence belong to Class (ii). One exception is Takayama andAkamatsu (2011) who also included global dispersion force by introducing mobile firmsand immobile consumers in each region. This model thus contains both local and globaldispersion force, i.e., of Class (iii).34

Other relevant models

In the NEG literature, a particularly important deviation from the canonical models is toconsider different transport cost structures by industry. For example, Fujita and Krugman(1995) included transport costs for (urban) differentiated products as well as land-intensive(rural) homogenous products. In the presence of rural goods that are costly to transport,the delivered price for such goods is lower in regions farther away from cities, whichgenerates a dispersion force. This is similar to the local dispersion force in that even asmall deviation from an urban agglomeration will decrease the price of rural goods and

33NEG models adopting transport costs that are not iceberg form are not studied in Akamatsu et al.(2018). But, it is known that they can also be classified according to the spatial scale of dispersion. Forexample, Ottaviano et al. (2002) and Tabuchi et al. (2005b), both of which adopt additive transport costs,belong essentially to Class (i) and Class (ii), respectively (see Akamatsu et al., 2018, §3.1).

34The social interactions model by Picard and Tabuchi (2013) with non-iceberg transport costs can beshown to belong to Class (iii) (see Akamatsu et al., 2018, §3.1).

16

increase the payoff of the deviant. However, the advantage of dispersion persists outsidethe agglomeration, i.e., it depends on the distance structure of the model. This type ofdispersion force has been shown to result in the formation of an industrial belt, a continuumof agglomeration associated with multiple atoms of agglomeration as demonstrated bythe simulations in Mori (1997) and Ikeda, Murota, Akamatsu and Takayama (2017). Theformal characterization of industrial belts, however, remains to be carried out.

Among the extant social-interactions models, some distinguish location incentivesbetween firms and consumers/workers unlike the canonical models discussed above (e.g.,Ogawa and Fujita, 1980; Fujita and Ogawa, 1982; Ota and Fujita, 1993; Lucas and Rossi-Hansberg, 2002). This distinction is especially crucial for explaining the location patternswithin a city, while it may be less relevant for the purpose of explaining the spatial patternof cities. At present, little formal results have been obtained regarding the spatial patternof cities that arise in these models (see Osawa, 2016, for the recent theoretical developmentin this direction.)

Other relevant models that were not covered so far include the spatial oligopoly modelsdesigned to explain the agglomeration of retail stores (e.g., Wolinsky, 1983; Dudey, 1990;Konishi, 2005). In these models, consumers have imperfect information on the typesand prices of goods sold by stores before they visit them. The greater the agglomerationof stores, the more likely it is that consumers will find their favorite commodities. Theconcentration of stores is explained by the market-size effect due to taste uncertaintyand/or lower price expectation. The dispersion force is global one given by the exogenousand spatially dispersed demand. Thus, these models are expected to behave similarlyto Class (i) models above, although no extensive analyses have been conducted in thisdirection (see Konishi, 2005, §5, for the discussion on the spacing of retail clusters).35

3.2 Diversity in city size

The most popular theoretical explanation of power law for city-size distribution at thispoint may be the random growth theory (e.g., Gabaix, 1999; Duranton, 2006, 2007; Rossi-Hansberg and Wright, 2007; Ioannides, §8.2, 2012) which postulates that the growth ratesof individual cities follows Gibrat’s law (Gibrat, 1931), i.e., they are independently andidentically distributed random variables.

This theory is highly compatible with systems-of-cities models. For example, in themodel by Rossi-Hansberg and Wright (2007), individual industries are subject to city-levelpositive externality from agglomeration, but do not benefit from colocation with otherindustries, so that the externality is industry-specific. Then, each city would specialize ina single industry in the presence of urban costs due to scarcity of land and the need forcommuting in a city. If the industry- (or city-) specific productivity growth rates satisfy the

35See Economides and Siow (1988) for a related model that explains the spacing of market areas in whichmarkets are formed due to matching externalities that arise in the exchange of consumption goods.

17

basic assumptions of random growth theory (including Gibrat’s law), the model generatesthe power law for city size distributions. It is a plausible explanation, as we have seenin Section 2.3 that the specialized cities are rather ubiquitous (refer to Figure 4 and thecorresponding discussion) despite the strong evidence for the hierarchy principle a laChristaller (1933).

A key implication of the random growth theory is that similar power laws hold for all(sufficiently large) random subsets of cities in a country, i.e., without any regard to spatialrelation among cities. Thus, this theory essentially denies the mutual dependence of sizeand spatial patterns of cities. But, as discussed in Section 2.2, Mori et al. (2019) have shownthat the similarity in power laws for city size distributions is much stronger among thecells in the spatial hierarchical partitions of cities that are consistent with the central placepatterns than among random subsets of cities, i.e., if city sizes were generated by a randomgrowth process.36,37

To account for the large diversity in city size observed in reality by the many-regionmodels described in Section 3.1, one needs to incorporate diversity in increasing returns(and/or that in transport costs). While Class (i) models with a global dispersion forcediscussed above can account for the formation of multiple cities, there is little variationin the sizes of cities to be realized in equilibrium, since each model has only one type ofincreasing returns.

There has been attempts of formalizing the central place thoery of Christaller (1933) byintroducing multiple industries subject to different degrees of increasing returns. The ini-tial formal attempt was made by Beckmann (1958). But, his model lacked microeconomicfoundation. Later the models with more explicit mechanisms were developed by Fujita,Krugman and Mori (1999b); Tabuchi and Thisse (2011) in the context of the NEG, andby Hsu (2012) in the context of spatial competition model. In these models, the differentdegrees of increasing returns among industries result in the different spatial frequenciesof agglomeration among industries.

The key to account for the diversity in city size in these models is the spatial coordi-nation of agglomerations among industries through inter-industry demand externalitiesthat arise from common consumers among industries. An industry subject to a largerincreasing returns agglomerate in a smaller number of cities that are farther apart. Whatis crucial is that these cities are chosen from the ones in which more ubiquitous industriessubject to smaller increasing returns are located. Consequently, larger cities are formed atthe location in which the coordination of a larger number of industries takes place. Thisspatial coordination of industries accounts for the positive correlation between the size,spacing and industrial diversity of a city as observed in reality (Sections 2.1 and 2.3).

36See Rozenfeld et al. (2008); Rybski and Ros (2013) for other evidence against Gibrat’s law for city sizes.37There still are possibilities to extend random growth models by adding spatial relations among cities,

thereby account for the spatial fractal structure of city systems in terms of power laws of city size distribu-tions. See, for example, Ioannides (2012, §8.2.5) for a review of related attempts.

18

In particular, Hsu (2012) proposed a unique spatial competition model with productdifferentiation and scale economies in production, and provided at this point the mostfar reaching formal explanation for the mutual dependence between spatial pattern andsize diversity of cities. When the distribution of scale economies in production of eachfirm (which is expressed in terms of the industry-specific fixed cost for production inhis model) is regularly varying, then his model replicates the power law for city sizedistribution (Section 2.2) together with the positive correlation between size and spacingof cities (Section 2.1), the power law for the number and the average size of choice cities ofindustries (Section 2.3), as well as the hierarchy principle observed in Japan (Section 2.3).

Davis and Dingel (2019) offer an alternative mechanism of spatial coordination amongindustries which in turn results in hierarchy principle and the diversity in city sizes in thecontext of a systems-of-cities model.38 Specifically, the hierarchy principle in this modelarises from vertical heterogeneity in skill level among workers and skill requirement byindustries together with inter-industry positive externality that is confined within the samecity. The mechanism underling the spatial coordination among industries in this modelis different from the central place models above. On the one hand, a small city attractsonly low skill industries and workers as it offers only small city-level agglomerationexternality. On the other hand, a large city attracts both high and low skill industries andworkers. High skilled have an incentive to live there, since the city offers a large city-level agglomeration externality and they can afford to live there. Although residentiallocations near the city center are occupied by high skilled, low skilled still can afford tolive in locations with low land rent (due to longer commuting) near the city fringe, whileenjoying the large city-level externality.

Alternatively, Desmet and Rossi-Hansberg (2009), Desmet and Rossi-Hansberg (2014),Desmet and Rossi-Hansberg (2015), Desmet et al. (2017) incorporated dynamic external-ities through endogenous innovation and spillover effects. These models are fundamen-tally different from all the models discussed so far in that the exogenous heterogeneityamong regions are essential for city formation, i.e., agglomerations do not form sponta-neously. The uneven distribution of mobile agents resulting from the exogenous regionalheterogeneity is magnified by the spillover effects over time. One exception in this strandof literature is Nagy (2017) who incorporated the same dynamic externalities into theNEG framework, so that his model is capable of explaining the spontaneous formation ofmultiple cities together with the diversity in city sizes. While this model has been appliedto replicate the evolution of the US cities in 19th century, the properties of agglomerationand dispersion in this model have not been formally analyzed.

38Rossi-Hansberg and Wright (2007) formulate a random-growth model by using the systems-of-citiesmodel in which cities are specialized in a single different industries, and power laws emerge for sizes ofthese cities.

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4 Concluding remarks

This paper reviewed the models which explain the mutual dependence of spatial patternand sizes of cities. A many-region geography with variations in interregional distanceis an essential feature of a model, if the spatial pattern of cities were the subject of thestudy. Naturally, there have been very few formal attempts that explicitly dealt with thishigh-dimensional problem until recently with notable exceptions by Hsu (2012).

A breakthrough has been brought about by Akamatsu et al. (2012) who proposedto focus on the racetrack economy which involves many regions with heterogeneousinterregional distances, while preserving symmetry among the regions. By utilizingthe discrete Fourier transformation, they have demonstrated that the spatial patterns ofagglomeration that aries in the NEG models in a many region setup can be formallyanalyzed to a large extent. The same group of researchers have also developed theframework for systematic numerical analysis on a many-region geography based onthe numerical bifurcation theory and group-theoretic bifucation theory (e.g., Ikeda, Akamatsuand Kono, 2012; Ikeda et al., 2017). Their numerical approach makes it possible toexplore asymmetric geography (e.g., the presence of edges and heterogeneity in regionaladvantages) as well as two-dimensional location space in a many-region setup.

In this paper, drawing largely from Akamatsu et al. (2018) which applied the analyticaltool developed by Akamatsu et al. (2012) to a wide variety of extant agglomeration models,I have reviewed the spatial pattern of cities and its relation to city sizes implied by thesemodels. But, Hsu (2012) continues to be the only tractable model that can account for thelarge diversity in city size in association with the observed spatial pattern of cities. Thus,much to be expected in the future development in this respect.

Finally, no models so far have been successful in integrating intra- and inter-city space.In the models aiming to explain intra-city spatial patterns, the location behavior of firmsand that of workers are typically distinguished, and land consumption and/or land inputsby firms together with commuting are considered (e.g., Fujita and Ogawa, 1982; Ota andFujita, 1993; Lucas and Rossi-Hansberg, 2002; Picard and Tabuchi, 2013). The modelsaiming to explain inter-city spatial patterns, on the contrary, typically ignore differentlocation incentives between firms and workers (all models discussed in this paper belongto this group). But, it is not trivial to integrate these two spatial scales in one model.

Some extant NEG models consider commuting and land consumption (e.g., Anas,2004; Murata and Thisse, 2005). But, such urban structure is by assumption confinedwithin a given region, and does not extend beyond a single region. As is discussed inSection 3.1, in a many-region geography with variations in interregional distance, thesemodels belong to Class (ii), which means that at most unimodal agglomeration forms.Although each region in these models has monocentric urban structure by assumption, andhence, it is tempted to be interpreted as a “city”, they can generate essentially at most one“true” city.

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To fully account for the spatial pattern of cities, the distinction between inside andoutside each city should also be endogenized.

21

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